Smoking Cessation Programs


a. Construct the confidence interval that could be used to test the claim in Exercise 5. What feature of the confidence interval leads to the same conclusion from Exercise 5?

Verified step by step guidance

1. Step 1: Identify the data and parameters needed to construct the confidence interval. Determine the sample mean (\( \bar{x} \)), sample size (\( n \)), standard deviation (\( s \) or \( \sigma \)), and the confidence level (e.g., 95%).
2. Step 2: Choose the appropriate formula for the confidence interval based on the data. If the population standard deviation (\( \sigma \)) is known, use the formula \( \bar{x} \pm Z \frac{\sigma}{\sqrt{n}} \). If \( \sigma \) is unknown, use \( \bar{x} \pm t \frac{s}{\sqrt{n}} \), where \( t \) is the critical value from the t-distribution.
3. Step 3: Calculate the margin of error using the formula \( Z \frac{\sigma}{\sqrt{n}} \) or \( t \frac{s}{\sqrt{n}} \), depending on whether \( \sigma \) is known or unknown. Ensure you use the correct critical value (\( Z \) or \( t \)) corresponding to the confidence level.
4. Step 4: Construct the confidence interval by adding and subtracting the margin of error from the sample mean (\( \bar{x} \)). The confidence interval will be \( [\bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error}] \).
5. Step 5: Interpret the confidence interval in the context of the claim from Exercise 5. If the confidence interval does not contain the value specified in the null hypothesis, it supports rejecting the null hypothesis. Conversely, if the interval contains the value, it supports failing to reject the null hypothesis.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95% or 99%. It provides an estimate of uncertainty around a sample mean or proportion, allowing researchers to infer about the population. The width of the interval reflects the variability in the data and the sample size; larger samples yield narrower intervals.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. The results of hypothesis tests can be supported by confidence intervals, as both approaches assess the likelihood of observing the data under the null hypothesis.

Margin of Error

The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It quantifies the uncertainty associated with the sample estimate, indicating how much the estimate could vary from the true population value. In the context of confidence intervals, the margin of error is half the width of the interval and is influenced by the sample size and variability in the data.